Dopant-induced stabilization of three-dimensional charge order in cuprates

Abstract

We investigate the microscopic mechanisms behind the stabilization of three-dimensional (3D) charge order by Pr doping in YBa$_2$Cu$_3$O$_7$ (YBCO7). Density-functional-theory calculations locate the lowest-energy Pr superlattices for both Ba- and Y-site substitution. In the Ba-site case, the smaller Pr ion pulls the surrounding atoms inward. This breathing-mode distortion pins charge-stripe walls to the Pr columns and forces them to align along the $c$ axis. Y-site Pr is larger than the host ion, produces an outward distortion, and fails to pin the stripes. Coarse-grained Monte-Carlo simulations show that the stripe correlation length rises in step with the structural correlation length of the Pr dopant as observed in prior experiments. We thus identify Ba-site substitution and dopant-induced lattice pinning as the key mechanism behind 3D charge order in Pr-doped YBCO7. This approach provides quantitative guidelines for engineering electronic orders through targeted ionic substitution.

Figures (14)

• Figure 1: DFT lowest-energy structure and dopant positions for (a) pure YBCO7, (b) Ba-site Pr-doped YBCO7, when Pr atoms substitute 1/6 of the Ba atoms, and (c) Y-site Pr-doped YBCO7, when Pr atoms substitute 1/3 of the Y atoms. The black boxes mark out the corresponding unit cell. (d-f) Spin-density isosurfaces of the ground states in (a-c), where light yellow and blue represent spin up and down separately. The isosurface level is $0.019\mu_B$/Å$^3$. The red dashed lines represent the domain walls of the stripe order in CuO$_2$ planes. All domain walls are along the $b$-axis. Different domain walls are stacked/aligned along the $c$-axis and display a 3D CO. (g-i) Spin-density isosurfaces of the lowest-energy spin excitations for each case in (d-f) (vertically aligned corresponding figures); corresponding energy increases compared to their ground state energies in (d-f) are listed in units of meV per planar Cu atom.
• Figure 2: Illustration of local structural distortions due to electronic orders or presence of dopants. All atomic displacements are in units of $10^{-3}$ Å. (a) Top view of one CuO$_2$ plane in pure YBCO7 with stripe order ($q_a=1/3$) as shown in Fig. \ref{['fig:DFT_charge_order']}(d). The isosurfaces shown are of the spin density with yellow and blue marking opposite signs. The red dashed line marks the electronic domain wall in one unit cell. The black rectangle is the unit cell of the domain wall pattern. The red arrows show the atomic displacements of planar oxygen atoms caused by the domain wall. (b) Top view of a CuO$_2$ plane when a Pr dopant (yellow ball) replaces the nearest neighbor Ba. The black arrows show the resulting atomic displacements. We have forced the Cu atoms to be non-magnetic to avoid additional electronic ordering effects (e.g., those shown in panel (a)). (c) The local structural distortion caused by a Pr dopant replacing the Y (with Cu magnetism suppressed).
• Figure 3: Illustration of a lattice configuration with a dopant and CO domain wall grids in the $ac$ plane. The crystal structure in the background is a side view of Fig. \ref{['fig:DFT_charge_order']}(b). Pr dopants (on Ba sites) are shown as blue diamonds while Ba atoms are marked as blue squares. The CO domain wall locations in Fig. \ref{['fig:DFT_charge_order']}(e) are marked by red crosses, and red circles represent other possible CO domain wall positions (domain walls are assumed straight in the $b$ direction). The black rectangle represents the unit cell in Fig. \ref{['fig:DFT_charge_order']}(b) containing one CuO$_2$ bilayer. We use even layer number $l$ to represent a lower layer in a unit cell, and odd $l$ to represent an upper layer (for dopants or domain walls). The interlayer interaction in the unit cell is denoted as "intra-bilayer", while the interaction across different unit cells is denoted as "inter-bilayer".
• Figure 4: Monte Carlo results for the dopant and domain walls in a $60\times60$ supercell. (a) A snapshot of the dopant positions in real space in the $ac$ plane at 136K, where white ($n_{il}=1$) and black ($n_{il}=0$) pixels represent Pr dopants and Ba atoms, respectively. The lower panel shows a zoomed-in view inside the yellow box, where the red rectangular marks out one DFT unit cell shown in Fig. \ref{['fig:DFT_charge_order']}(b). (b) The averaged squared magnitude of the Fourier transform of the dopant configurations averaged over 1,000 snapshots. (c) Snapshot of the CO domain wall positions in real space in the $ac$ plane at 50K, where white/black pixel represents a position with/without a CO domain wall ($\tilde{n}_{il}$ is 1/0). The dopant alignment in (a) is used for this simulation. (d) The averaged squared magnitude of the Fourier transform of the $\tilde{n}_{il}$ following the recipe in (b). (e) and (f): The peak intensities along the in-plane and out-of-plane direction marked by the red arrows in (d). Blue dots show the computed Monte Carlo data, while the red curves are Lorentzian fits using a simple least-squares fit.
• Figure 5: Convergence of correlation lengths with respect to supercell size. (a) Dopant in-plane (IP) correlation length (CL) as a function of temperature, where the unit of length is the primitive cell lattice constant $a\approx3.8$Å. Different symbols represent results from different $L\times L$ supercell sizes marked as $L*L$. The error bars of the dopant CLs are the standard deviation of the CLs based on 100 independent MC simulations with different random initial guesses. (b) Dopant out-of-plane (OOP) CL, presented in the same manner as (a). The length unit is half of the primitive cell lattice constant $c/2\approx5.9$Å.